[서평]「Component Software」 - Beyond Object-Oriented Programming -

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Classical Descriptive Set Theory

Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

In this chapter we will present the basic concepts of term rewriting that are needed in this book. More details on term rewriting, its applications, and related subjects can be found in the textbook of Baader and Nipkow [BN98]. Readers versed in German are also referred to the textbooks of Avenhaus [Ave95], Bundgen [Bun98], and Drosten [Dro89]. Moreover, there are several survey articles [HO80, DJ90, Klo92, Pla93] that can also be consulted.

Term Rewriting Systems

Some readers know to play the game of nim well, fewer play a perfect annihilation game, and nobody knows whether there exists an opening move in chess that will guarantee a win for white. These games and many more, belong to the family of combinatorial games, by which we mean the set of all two-player perfect-information games without chance moves and with outcomes lose or win (and sometimes: dynamic tie). The motivation for ONAG may have been, and perhaps was-and I would like to think that it was-the attempt to bridge the theory gap between nim-like and chess-like games. Why is there a gap? Every combinatorial game can be described as a directed graph called game-graph, whose vertices are the game positions, and (u, v) is a directed edge if and only if there is a move from position u to position v. Denote by N the set of all positions from which the Next (first) player can force a win; by P the set of all positions from which the Previous (second) player can force a win; and by T the set of all (dynamic) Tie positions, which are positions from which no player can force a win and therefore both can avoid losing. In an acyclic game-graph there cannot be any tie positions. The N, P, T classification of any game graph R = (V, E) can be determined in 0(\V\ + \E\) steps [8]. For both nim and chess, a finite game-graph can be constructed and the N, P, T classification can be determined. So both games are solvable in principle. If we play nim with n piles, each pile containing at most k tokens, then the game-graph contains (k + \) vertices. Suppose that in (generalized) chess played on an « X « board there are k different pieces. If k is about n/2, then the game-graph of chess contains O (2") vertices. So both game-graphs have exponentially many vertices, and thus both games appear intractable in the usual sense of computational complexity [1, Chapter 10], [14, Chapter 9], namely a computation appears to be required which is asymptotically exponential. From a computational efficiency standpoint, the essential difference between nim and chess is that nim can be viewed as a disjunctive compound (sum) of independent games, namely the individual piles. A disjunctive

http://ieeexplore.ieee.org/iel5/5/31266/01455274.pdf

On numbers and games

The powerset construction is a standard method for converting a nondeter- ministic automaton into a deterministic one recognizing the same language. In this paper, we lift the powerset construction from automata to the more general framework of coal- gebras with structured state spaces. Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems. An endofunctor F determines both the type of systems (F-coalgebras) and a notion of behavioural equivalence (�F) amongst them. Many types of transition systems and their equivalences can be captured by a functor F. For example, for deterministic automata the derived equivalence is language equivalence, while for non-deterministic automata it is ordinary bisimilarity. We give several examples of applications of our generalized determinization construc- tion, including partial Mealy machines, (structured) Moore automata, Rabin probabilistic automata, and, somewhat surprisingly, even pushdown automata. To further witness the generality of the approach we show how to characterize coalgebraically several equivalences which have been object of interest in the concurrency community, such as failure or ready semantics.

/pdf/generalizing-determinization-from-automata-to-coalgebras-27uexpcxfw.pdf

Generalizing determinization from automata to coalgebras

NetKAT is a domain-specific language and logic for specifying and verifying network packet-processing functions. It consists of Kleene algebra with tests (KAT) augmented with primitives for testing and modifying packet headers and encoding network topologies. Previous work developed the design of the language and its standard semantics, proved the soundness and completeness of the logic, defined a PSPACE algorithm for deciding equivalence, and presented several practical applications. This paper develops the coalgebraic theory of NetKAT, including a specialized version of the Brzozowski derivative, and presents a new efficient algorithm for deciding the equational theory using bisimulation. The coalgebraic structure admits an efficient sparse representation that results in a significant reduction in the size of the state space. We discuss the details of our implementation and optimizations that exploit NetKAT's equational axioms and coalgebraic structure to yield significantly improved performance. We present results from experiments demonstrating that our tool is competitive with state-of-the-art tools on several benchmarks including all-pairs connectivity, loop-freedom, and translation validation.

/pdf/a-coalgebraic-decision-procedure-for-netkat-x1vt970weg.pdf

A Coalgebraic Decision Procedure for NetKAT

This paper presents a new language for network programming based on a probabilistic semantics. We extend the NetKATlanguage with new primitives for expressing probabilistic behaviors and enrich the semantics from one based on deterministic functions to one based on measurable functions on sets of packet histories. We establish fundamental properties of the semantics, prove that it is a conservative extension of the deterministic semantics, show that it satisfies a number of natural equations, and develop a notion of approximation. We present case studies that show how the language can be used to model a diverse collection of scenarios drawn from real-world networks.

Probabilistic NetKAT

In this paper, we present a systematic way of deriving (1) languages of (gener- alised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of systems. This generalizes both the results of Kleene (on regular languages and de- terministic finite automata) and Milner (on regular behaviours and finite labelled transition systems), and includes many other systems such as Mealy and Moore machines.

/pdf/non-deterministic-kleene-coalgebras-55f2m2ullc.pdf

Non-Deterministic Kleene Coalgebras

Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems. An endofunctor $F$ determines both the type of systems ($F$-coalgebras) and a notion of behavioral
equivalence ($\sim_F$) amongst them. Many types of transition systems and their equivalences can be captured by a functor $F$. For example, for deterministic automata the derived equivalence is language equivalence, while for non-deterministic automata it is ordinary bisimilarity. The powerset construction is a standard method for converting a nondeterministic automaton into an equivalent deterministic one as far as language is concerned. In this paper, we lift the powerset construction on automata to the more general framework of coalgebras with structured state spaces. Examples of applications include partial Mealy machines, (structured) Moore automata, and Rabin probabilistic automata.

/pdf/generalizing-the-powerset-construction-coalgebraically-2d6pfx9qlp.pdf

Alexandra Silva

Papers

Generalizing determinization from automata to coalgebras

A Coalgebraic Decision Procedure for NetKAT

Probabilistic NetKAT

Non-Deterministic Kleene Coalgebras

Generalizing the powerset construction, coalgebraically